Surface wave inversion by wave field reconstruction and wave equation inversion

We present an inversion scheme for surface waves that jointly inverts point-receiver data for both a densely reconstructed wave field and an estimate of seismic velocities. The formulation is posed as a partial differential equation (PDE) constrained inverse problem. We use the dispersive Helmholtz equation to approximate the far-field behavior of surface waves as two dimensional wave propagation through a phase velocity map. The Helmholtz equation does not accurately describe surface wave propagation near sources so we mask source regions from the PDE constraint. This leads to excellent wave field reconstruction and medium velocity estimates. The new theory and algorithm are supported by a numerical example with simulated elastic data. The phase velocity model obtained is verified by frequency-wavenumber dispersion analysis.